A linear algebraic group of type \(F_4\)  is the group of
automorphisms of a so-called Albert \(k\) -algebra \(\mathbf {A}\)  (see
[1]} 17.5). We have \(|H^1(k,F_4(\overline{k}))|=3\)  and \(\operatorname{Aut}F_4 \simeq F_4\) , whence \(H^1(k,F_4(\overline{k}))\)  classifies the
real forms of \(F_4\)  up to \(k\) -isomorphism. A linear differential
equation \(\mathcal {L}(Y)=0\)  defined over \(K\) , with differential Galois
group \(F_4\) , has three Picard-Vessiot extensions, up to
\(K\) -differential isomorphism, and each of the three real forms of
\(F_4\)  appears as the differential Galois group of one of the
Picard-Vessiot extensions.
